liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Numerical Solution of the Cauchy Problem for the Helmholtz Equation
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.ORCID iD: 0000-0002-2681-8965
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. University of Rwanda.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2014 (English)Report (Other academic)
Abstract [en]

The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.

Place, publisher, year, edition, pages
Linköping University Electronic Press, 2014. , 16 p.
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2014:04
Keyword [en]
Helmholtz equation, Cauchy Problem, Ill-Posed, Regularization, Numerical Methods.
National Category
Computational Mathematics Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-105707ISRN: LiTH-MAT-R--2014/04--SEOAI: oai:DiVA.org:liu-105707DiVA: diva2:709791
Available from: 2014-04-03 Created: 2014-04-03 Last updated: 2014-04-11Bibliographically approved
In thesis
1. Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation
Open this publication in new window or tab >>Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The inverse problem of reconstructing the acoustic, or electromagnetic, field from inexact measurements on a part of the boundary of a domain is important in applications, for instance for detecting the source of acoustic noise. The governing equation for the applications we consider is the Helmholtz equation. More precisely, in this thesis we study the case where Cauchy data is available on a part of the boundary and we seek to recover the solution in the whole domain. The problem is ill-posed in the sense that small errors in the Cauchy data may lead to large errors in the recovered solution. Thus special regularization methods that restore the stability with respect to measurements errors are used.

In the thesis, we focus on iterative methods for solving the Cauchy problem. The methods are based on solving a sequence of well-posed boundary value problems. The specific choices for the boundary conditions used are selected in such a way that the sequence of solutions converges to the solution for the original Cauchy problem. For the iterative methods to converge, it is important that a certain bilinear form, associated with the boundary value problem, is positive definite. This is sometimes not the case for problems with a high wave number.

The main focus of our research is to study certain modifications to the problem that restore positive definiteness to the associated bilinear form. First we add an artificial interior boundary inside the domain together with a jump condition that includes a parameter μ. We have shown by selecting an appropriate interior boundary and sufficiently large value for μ, we get a convergent iterative regularization method. We have proved the convergence of this method. This method converges slowly. We have therefore developed two conjugate gradient type methods and achieved much faster convergence. Finally, we have attempted to reduce the size of the computational domain by solving well–posed problems only in a strip between the outer and inner boundaries. We demonstrate that by alternating between Robin and Dirichlet conditions on the interior boundary, we can get a convergent iterative regularization method. Numerical experiments are used to illustrate the performance of the  methods suggested.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2014. 12 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1593
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-105879 (URN)10.3384/diss.diva-105879 (DOI)978-91-7519-350-2 (ISBN)
Public defence
2014-05-09, ACAS, A–huset, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Supervisors
Note

An invalid ISRN (LIU-TEK-LIC-2012:15) is stated on page 2. The ISRN belongs to the Licentiate thesis, published in 2012.

Available from: 2014-04-11 Created: 2014-04-11 Last updated: 2014-04-11Bibliographically approved

Open Access in DiVA

Numerical Solution of the Cauchy Problem for the Helmholtz Equation(1162 kB)561 downloads
File information
File name FULLTEXT01.pdfFile size 1162 kBChecksum SHA-512
d6b991c91ec229f9124a3062352e8b879ab945473607d659e1a3b46e6c43669e573fdd6af9585690864ea7a43dcb20acb82fdd13e187a410cb1d57bda73cee73
Type fulltextMimetype application/pdf

Authority records BETA

Berntsson, FredrikKozlov, Vladimir A.Mpinganzima, LydieTuresson, Bengt-Ove

Search in DiVA

By author/editor
Berntsson, FredrikKozlov, Vladimir A.Mpinganzima, LydieTuresson, Bengt-Ove
By organisation
Computational MathematicsThe Institute of TechnologyMathematics and Applied Mathematics
Computational MathematicsMathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 561 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 297 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf